Purity for flat cohomology

• 成果类型: Article
• 署名作者: Cesnavicius, Kestutis; Scholze, Peter
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2024.199.1.2
• 发表日期: 2024
• 页码: 51-180
• 关键词: dieudonne theory; projectivity; lefschetz; PROPERTY; HOMOLOGY; rings

摘要

We establish the flat cohomology version of the Gabber-Thomason purity for & eacute;tale cohomology: for a complete intersection Noetherian local ring (R,m) and a commutative, finite, flat R-group G, the flat cohomology H-m(i)(R,G) vanishes for for i <= dim(R). For small i, this settles conjectures of Gabber that extend the Grothendieck-Lefschetz theorem and give purity for the Brauer group for schemes with complete intersection singularities. For the proof, we reduce to a flat purity statement for perfectoid rings, establish p-complete arc descent for flat cohomology of perfectoids, and then relate to coherent cohomology of AInf via prismatic Dieudonn & eacute; theory. We also present an algebraic version of tilting for & eacute;tale cohomology, use it to reprove the Gabber-Thomason purity, and exhibit general properties of fppf cohomology of (animated) rings with finite, locally free group scheme coefficients, such as excision, agreement with fpqc cohomology, and continuity.